Phy Sci 119a
Orbits, Newton’s Laws
Opening music; Franz Haydn (1732-1809): Symphony No. 94, the "Surprise". Date of first performance, 1791. The performance date is near the time of the Cavendish Experiment that established the value of G, the gravitational constant. That piece was written just after Herschel published his first big catalogue of galaxies (1785) and a few years before Haydn visited Herschel to see the world's largest telescope.
Opening art: The Death of Marat (1893) by Jacques-Louis David (1758-1825). David took an active part in the French Revolution. Painted in the year Louis XVI was beheaded, his greatest picture commenorates a popular hero of the Revolution murdered in his bath. Marat had a painful skin condition that required him to work while soaking in a tub. One day, a young woman, Charlotte Corday, burst in with a personal petition and plunged a kife into his chest while he read it. Now located in the Musee Royaux des Beaux Arts de Belgique, Brussels.
Opening costume: Empire-style dress, shown in the painting Comtesse Daru (1810)
by Jacques-Louis David. In the year of the Cavendish experiment (1796), Napoleon was made major general, winning a major victory over the Austrians in Italy. With the establishment of the Empire by 1804, Napoleon promoted more elaborate styles requiring expensive and ornate fabrics with which he attempted to recreate the elegance of the regime ancien. A bustle like construction placed at the back of the waistline gave the body a forward slant, known as the "Grecian bend".
Closing music: Vivaldi (1646-1741): "Spring" from the Four Seasons , first performed in 1725. The piece was first performed just before Newton died, and so can be remembered as the end his era.
Closing art; The Ancient of Days (1794). This is the frontispiece of Europe, A Prophesy. A reclusive visionary, the artist, William Blake (1757-1827) produced illustrated texts after the fashion of medieval illuminated manuscripts. This watercolor borrows its symbolism from representations of the Lord as Architect, creator of the Universe. Blake regarded the Enlightenment's worship of reason as ultimately destructive, as it stifles inspiration. To Blake, the "inner eye" of the artist was all-important; he felt no need to observe the visible world around him. This workis a critique of the scientific fervor of the time of Cavendish.
Literature: Coleridge wrote "The Rime of the Ancient Mariner" in 1798, as noted in the previous lecture. "Tintern Abbey" by William Wordsworth (1770-1850), appeared in the same collection of poetry in the same year. Wordsworth thought of the Universe as a "living whole" and rejected science and mathematics for their rationalism.
Cavendish and G
Determination of Solar System Parameters
I briefly described the Cavendish experiment (1796), which I will do in the lecture on Monday. Two small balls are placed on a rod to make a small dumbbell. The rod is suspended on a cord inside an air-tight, thin box (to remove air currents that could disturb the experiment). Two larger balls are placed next to the two small balls, but outside the box, on a pivoting rod. The big balls attract the small ones with an equal and opposite force, and the small balls reach an equilibrium state at rest, after sitting overnight. The two large balls, on the pivoting bar, are quickly rotated through just under 180o. The force on the small balls is thus changed by F=-2GmM/r2. The attraction of the small balls to the large balls is recorded on the wall by marking the position of the laser spot reflected off of a mirror fastened to the small rod. Initially, little change is noticed, but, as time passes, each successive dot is further and further from the previous one. (The dumbbell inside the box is on the end of a ribbon, so that as the dumbbell rotates, the tension builds in the ribbon, stops the gravity induced motion, and sends the balls the other direction. The system oscillates to a new equilibrium. The laser will thus trace a line on the wall, allowing one to repeat the measurement several times.)The acceleration of the dots can be converted to find the acceleration of the small balls, using geometry. Since F=ma, and we can measure m, the determination of a gives G. Then, the above equation can be solved for G, all other quantities being measurable in the lab. The value of G so derived seems to be constant through time and space in this Universe. It gives consistent answers, where it can be checked, over distances of dozens of astronomical units. If G varied in time, then models of stars done in computers would not explain the observations, as gravity is an important component of the equations that describe the structure of stars.
The force that attracts the balls is algebraically the same as the force acting between Earth and a falling ball, or, as Newton showed, between the Earth and the Moon. The acceleration of the Moon towards Earth is v2/r (from Huygen’s rule). Using the distance to the Moon and the velocity of the Moon in its orbit, one finds that a=0.26 cm/sec2. The acceleration is along the Earth-Moon line. If one forms the ratio (RE2/rMoon2), it is seen to be equal to the ratio of the acceleration of the Moon toward Earth and the acceleration of the ball toward Earth (980 cm/sec2). Since F=ma=-GmM/r2 according to Newton’s inverse square law of gravity, this is exactly what would be predicted.
Thus the law of gravity follows the same equation over separations of 10 cm (the Cavendish experiment), 6.4x108 cm (the radius of the Earth, hence the distance of the puck we used last Wednesday from the center of the Earth), and 3.8x1011cm (the distance to the moon). Using binary stars, one can show that the law applies over distances of 1014 cm (several astronomical units). However, beyond that, it cannot be tested. It is assumed to hold over the entire Universe. Numerous experiments have shown that the law is independent of the shape of a body, or the type of material (wood, silver, lead, etc.). The type of mass or energy does not matter. “Dark” matter, black holes, things made of protons and neutrons, energy in any form (E=mc2): all follow the same law of gravity with the same gravitational constant. Only the mass (or mass equivalent) of the attracting bodies matters.
The fact that the dropping ball follows the law of gravity at the surface of Earth, proves a final fact about gravity. The mass of a body can be mathematically placed at the center of mass. There are no effects due to extended bodies, as far as we are concerned in this course. These statements are strictly true for objects with uniform density distributions. The Earth is denser at the core than at the center, but is roughly symmetrical.
Without knowledge of G, even the use of Newton’s laws cannot tell us the key parameters of our system: the mass of the Sun, the mass of the Earth, the mass of the Moon, the radius of these bodies and the distance of these bodies from Earth. I discussed the estimation of these quantities independently of Newton’s Laws of motion and of gravity.
First, the radius of the Earth. The first person known to have done this was Erastothenes, around 200 BC. He noted that the Sun could be seen from the bottom of deep wells, drilled directly toward the center of Earth, in a town in Syene, Egypt. From his own home in Alexandria, this phenomenon never occurred. The Sun was seen, as it crossed the meridian (the plane perpendicular to the equator passing overhead at a given site), to be 70 from the central line of local wells. Draw the line through the centers of wells (verticals) at the two sites. Measure the distance between the two cities. The line to the Sun from Alexandria is essentially parallel to the well centerline at Syene, since the Sun is so far away. By construction, the centerlines of the wells make an angle of 70. We then have the classic skinny triangle. The distance between the cities divided by the radius of the Earth must yield the angle measured by Erastothenes. Therefore, the radius can be derived. He found RE~6660 km. (He used units that we do not use today and that we do not exactly understand, but the best guess at a conversion is used here.) The value can be compared to the actual value we know today, RE=6378, accurate to about 0.5%.
Next, consider the size of the Moon. (The argument is due to Aristarchus of Samos.) When the Earth eclipses the moon, the moon travels through the shadow of Earth. The angle subtended by the point of ingress into shadow and the point of egress can be measured. The angular size of the moon fits 3.7 times into that angle. But, because the Sun’s rays are almost parallel, the projection of the shadow at the distance of the moon is essentially the diameter of Earth. Therefore, the moon is 1/3.7 times the size of the Earth. The true value of RM is 1738 km. Once Erastothenes measured the radius of the Earth, the Greeks would have computed RM =1800 km.
Now consider the distance to the moon, rm. The time it takes for the moon to go into and emerge from the Earth’s shadow (3.5 hours) divided into the size of the projected shadow give the velocity of the Moon in its orbit, 1 km/sec. The period is known, about 28 days, and P=2rm/vm. Therefore, the distance to Moon follows, rm=384,400 km/sec.
Aristarchus realized that when the moon was half full, the line from the Moon to the Sun (a) makes a right angle to the Earth/Moon line (b). Another right triangle is formed by the Earth Sun line (c) and the line between the Moon and the Earth when the Moon is straight overhead (d). The difference is 0.14 degrees. By geometry, that angle is the same as the angle between lines a) and c), at the Sun,, 0.14 degrees. By the skinny triangle, rm=rsxTherefore, rS=1.5x108 km or 1.5x1013cm (defined as one astronomical unit). Note that must be divided by the number of degrees per radian (57) to do this calculation.
With the distance from the Sun to the Earth known (1 astronomical unit, or 1 AU), the velocity of the Earth around the Sun is known, since we know the period, 1 year. P=2rS/vS, so vE=30 km/sec.
Finally, Aristarchus of Samos noted that the Moon and the Sun are exactly the same angular diameter (as shown by the fact that in eclipse, the moon just exactly covers the Sun.) Therefore, Rm/rm=RS/rS and the Solar radius is RS=7x1010 cm.
To get the masses of the three bodies, the Greeks would have had to assume the density of each body and multiply by the volumes (known, since they knew the radii of the bodies.) A reasonable guess might have been 2 g/cm3 for Earth, since about half the Earth they knew about was water, at the surface (1g/cm3) and half sand or soil, 2-3 g/cm3. Since the ocean everywhere had a bottom, they might even have guessed that the true value was probably a little higher. They might then have guessed that the average density of the other bodies was the same. For the Moon, that calculation gives a mass (density times volume) of 5.5x1025 grams. For the Earth, 2.7x1026 grams. For the Sun, 3.5x1033 grams. The true values are, respectively, 7.3x1025, 6.0x1027 and 2x1033 grams.
From Newton’s laws and the above techniques, Newton could thus have estimated G, using the equation for orbital velocity for the Earth/Sun system or for the Earth Moon systems and the mass estimates above. He would have gotten values accurate to about a factor of 2, implying that G was a constant over a factor of 40,000 in distance. He would have been amazed! (I am not sure he actually did this.)